A GCSE-level instructional video (Third Space Learning) by Paul explaining how to use and rearrange Pythagoras’ theorem to find a shorter side of a right-angled triangle. The video demonstrates algebraic rearrangement, works through example exam questions (including numeric substitution, squaring, square roots, rounding to 1 decimal place), highlights common mistakes (subtracting in the wrong order), shows marking allocation, and points viewers to additional resources and tutoring in the description.
Chapters
Hi, this is Paul from Third Space Learning and today we're going to look at using Pythagoras theorem to work out the lengths of one of the shorter sides. Now make sure you've watched our video on finding the longer side using Pythagoras Theorem. You'll find the link in the description below. As usual, I'll be using our online classroom to go through the topic, look at some questions and then at the end of the video you'll find links to our free resources, practice questions and and information on our one to one in school tutoring. Okay, let's look at Pythagoras Theorem to find the shortest side of a right angled triangle. Grab a pen and let's dive in. So we remember that Pythagoras theorem is used to work out the lengths of right angle triangles. A squared plus B squared equals C squared. Now in order to work out one of those shorter sides, all we have to do A squared equals C squared minus B squared. So let's have a look at that in a little more detail. Here's a right angle triangle. We've got a hypotenuse length, the longest side of five and another length of four. So let's square these. We can square five. Five times five is 25 and we can square four. Four times four is 16. Now let's subtract these 25, subtract 16 is nine. And what we worked out there is the square of that missing side. And then to work out the actual missing side, all we can do is take the square root of it. So we've got A squared equals C squared, subtract B squared. Let's see how this looks on a real exam question. Below we have a right angled triangle. So we can see that we've got two of the sides of the right angled triangle and we want to work out the third. So we can use Pythagoras. And Pythagoras theorem says A squared plus B squared equals C squared. Okay, let's get the triangle labeled. Well, the side opposite the right angle is the longest side. That's C. And then A and B, it doesn't really matter. So let's have A as the side we don't know and B as the side we do. Now, because we want to work out A, let's rearrange Pythagoras theorem. So let's subtract B squared from both sides of Pythagoras theorem. Well, if we do that, what we're left with is A squared on the left, C squared, take away B squared on the right. And this tells me that we're going to square the longest side, we're going to square one of the shorter sides and we're going to subtract them. So let's put the numbers in. Well, A is X, so let's replace that X squared. C is 13, so that's 13 squared. And B is 5. So let's take away the 5 squared. 13 squared is 169, 5 squared is 25. And then 169 take away 25 is 144. So that's X squared. We want to work out the length X. So let's take the square root of that 144 square rooted, which is 12 and the all important units are centimeters. Now a common mistake people make here is by taking away the 13 and the five in the wrong order, we need to make sure we always take away the shorter side from the longer side because if we don't we'll end up with a negative length, which is impossible. Quick check to make sure our answer is sensible. Now we wanted to work out one of the shorter sides. So 12 should be shorter than than the longest side, which is 13. And it is. So that looks great to me and we can pop that on the answer line. 12cm. Let's see where the marks come from. So it's a two mark question. One mark is for correctly substituting the numbers into Pythagoras. And the second mark comes from the final answer. Here's a second exam question. Pause the video and have a go. See if you can work out what Y is. Okay, let's have a go. So triangle def is right angled and we can see the right angle. Here we know two of the sides, five and seven and we want to find Y correct one decimal place. Okay, so this means we can use Pythagoras. So A squared plus B squared equals C squared. Let's label our triangle. Now C is always the longest side and it's the side opposite that right angle. So, so that's C there, 7 and A and B doesn't matter. So let's have A is the side we don't know and B is the side we do. Now because we want to work out A, we need to rearrange Pythagoras. So let's take B squared away from both sides. So we're left with A squared is equal to C squared. Take away B squared. Okay, let's substitute in the values so we've Got A, which is Y. So that's Y squared. C is 7, so that's 7 squared. Subtract B which is 5, 5 squared. Now because it says to one decimal place, the answer is probably going to be a decimal. So let's just pop this straight into the calculator. So 7 squared takes. Take away 5 squared is 24. Let's look at why. Let's take the square root of 24 which is equal to 4.89897. Always a good idea to write down a good number of decimal places. Now it says to one decimal place. So let's round it. The 8 is in the first decimal place. So draw a line to the right. Now we've got a nine there. So nine is bigger than five. So we're going to round up. So My answer is 4.9, 4.9 cm, those all important units. Let's do a quick check to make sure the answer is sensible. So we wanted to work out one of the shorter sides. So 4.9 should be smaller than the longest side of the triangle, the hypotenuse, which is. Which is seven and it is. So that looks great. So let's put that on the answer line. 4.9 centimeters. Thanks so much for watching the video. Check out the description below for loads more information on this topic and for access to our library of free online resources and loads of information on our one to one tuition. Don't forget to like and subscribe to stay up to date with all the latest third space learning videos.
